1 / 5

Methods for Pseudo Invariants

Methods for Pseudo Invariants. Decoupled Damped Systems. Restore “Determinant=1” by scaling the transfer map by the damping factor where D is the determinant of the linear map of the decoupled damped system. Coupled Systems.

tallis
Download Presentation

Methods for Pseudo Invariants

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Methods for Pseudo Invariants Decoupled Damped Systems • Restore “Determinant=1” by scaling the transfer map by the damping factor where D is the determinant of the linear map of the decoupled damped system. Coupled Systems • First, decouple the system to 2 x 2 blocks by taking real and imaginary parts of eigen vectors. The Normal Form method can be used for analysis of nonlinear behavior. M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  2. Tracking a Decoupled Damped System- Quadrupole Cooling Cells - Tracking via the damped transfer map Tracking via the determinant 1 restored transfer map of the damped system M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  3. Normal Form Theory Goal: perform a nonlinear change of variables such that the motion in the new variable pairs is rotationally invariant: If the map is symplectic, this means circles. If the map is damped, we obtain logarithmic spirals. Advantage: Tune with amplitude is trivial to compute, since each iteration of the map corresponds to the same angle advance. Other Advantages: - Provides pseudo invariants the quality of which allows conclusions about the map; - sensitive to resonances, allows efficient study of resonances. M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  4. Tracking a Coupled Non-Damped System- sFOFO 2.75m Cells from Study II - Tracking in Regular (x-x’) Coordinates 1-20 Cells 101-120 Cells Tracking in Normal Form Coordinates 1-300 Cells M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

  5. Tracking a Coupled Damped System- sFOFO 2.75m Cooling Cells from Study II - Tracking in Regular (x-x’) Coordinates 1-20 Cells 101-120 Cells Tracking in Normal Form Coordinates 1-300 Cells 41-60 Cells M. Berz Emittance Exchange Workshop at LBNL, October 3-19, 2001

More Related